Optimality

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164 D. M. Dabrowska so that in both cases it is enough to consider the determinants for ordered sequences s = (s1, . . . ,sm), s1 < s2 < . . . < sm of points in (0, τ] m . For any such sequence s, the matrix dm(s) has a simple pattern: ⎛ c(s1) ⎜ c(s1) ¯dm(s) ⎜ = ⎜ c(s1) ⎜ ⎝ . c(s1) c(s2) c(s2) c(s1) c(s2) c(s3) . . . . . . . . . c(s1) c(s2) c(s3) . ⎞ ⎟ . ⎟ ⎠ c(s1) c(s2) c(s3) . . . c(sm) We have ¯ dm(s) = A T mCm(s)Am where Cm(s) is a diagonal matrix of increments Cm(s) = diag [c(s1)−c(s0), c(s2)−c(s1), . . . c(sm)−c(sm−1)], (c(s0) = 0, s0 = 0) and Am is an upper triangular matrix ⎛ 1 ⎜ 0 ⎜ Am = ⎜ . ⎜ . ⎝ 0 1 1 0 . . . 1 . . . 1 . . . 1 1 1 . 1 0 0 . . . 0 1 To see this it is enough to note that Brownian motion forms a process with independent increments, and the kernel k(s, t) = c(s∧t) is the covariance function of a time transformed Brownian motion. Apparently, det Am = 1. Therefore and det ¯ dm(s) = ⎞ ⎟ . ⎟ ⎠ m� [c(sj)−c(sj−1)] det ¯ Dm(t, u;s) = det ¯ dm(s)[c(t∧u)−Um(t;s)[ ¯ dm(s)] −1 Vm(s;u)] j=1 = det ¯ dm(s)[c(t∧u)−Um(t;s)A −1 m C −1 m (s)(A T m) −1 Vm(s;u)]. The inverse A−1 m is given by Jordan matrix ⎛ 1 −1 0 . . . 0 0 ⎜ 0 1 −1 . . . 0 0 A −1 m = ⎜ ⎝ and a straightforward multiplication yields det ¯ Dm(t, u;s) = c(t∧u) − × . . . . 0 0 0 . . . 1 −1 0 0 0 . . . 0 1 m� [c(sj)−c(sj−1)] j=1 ⎞ ⎟ ⎠ m� [c(t∧si)−c(t∧si−1)][c(u∧si)−c(u∧si−1)] i=1 m� j=1,j�=i [c(sj)−c(sj−1)].
Semiparametric transformation models 165 By noting that the i-th summand is zero whenever t∧u < si−1 and using induction on m, it is easy to verify that for t≤u the determinant reduces to the sum det ¯ Dm(t, u;s) = 1(t≤u
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Institute of Mathematical Statistic
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Institute of Mathematical Statistic
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iv Contents Copulas and Decoupling
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vi Finally, thanks go the Statistic
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SCIENTIFIC PROGRAM The Second Erich
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x Recent Advances in Longitudinal D
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The Second Lehmann Symposium—Opti
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xiv Richard Davis Colorado State Un
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xvi Matthias Matheas Rice Universit
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xviii Hui Zhao University of Texas
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IMS Lecture Notes-Monograph Series
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Proof. The maximum LR test rejects
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4. Location-scale families On likel
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6. Conclusions On likelihood ratio
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Student’s t-test for scale mixtur
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Optimality in multiple testing 17 w
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Optimality in multiple testing 19 I
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power 0.02 0.03 0.04 0.05 0.06 0.07
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Optimality in multiple testing 23 i
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Optimality in multiple testing 25 (
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Optimality in multiple testing 27 M
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6. Summary Optimality in multiple t
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Optimality in multiple testing 31 [
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On the false discovery proportion 3
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≤ N� � N� P m=1 On the fals
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Since j≤ s, it follows that On th
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0.025 0.02 0.015 0.01 0.005 On the
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Massive multiple hypotheses testing
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P value EDF qdf 0.0 0.2 0.4 0.6 0.8
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Proof. See Appendix. Massive multip
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X1 Massive multiple hypotheses test
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0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4
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Then π0α ∗ cal π0α ∗ cal +
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Frequentist statistics: theory of i
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2.3. Failure and confirmation Frequ
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3.1. Types of null hypothesis Frequ
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together with the probabilistic ass
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Statistical models: problem of spec
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Page 153 and 154: Semiparametric transformation model
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Page 205 and 206: Copulas, information, dependence an
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Regression trees 215 of the tree. T
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Regression trees 217 GUIDE methods
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Regression trees 219 and E appear a
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Regression trees 221 Table 3 Result
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Solder =thick 2.5 Regression trees
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Observed values 0 10 20 30 40 50 60
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Regression trees 227 effects and in
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On Competing risk and degradation p
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Restricted estimation in competing
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9. Conclusion 0.0 0.1 0.2 0.3 0.4 0
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2.1. Maximin efficiency robust test
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Robust tests for genetic associatio
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1 . . . 1 i . . . . . . R j . . . O
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Optimal sampling strategies 269 as
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Optimal sampling strategies 271 γ1
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Optimal sampling strategies 273 Def
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Optimal sampling strategies 275 Usi
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Optimal sampling strategies 277 Def
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optimal 1 2 3 4 leaf sets 5 6 (leaf
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6. Related work Optimal sampling st
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Optimal sampling strategies 283 Cla
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Optimal sampling strategies 285 Sin
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Optimal sampling strategies 287 µ
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Optimal sampling strategies 289 on
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Linear prediction after model selec
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y the P× 1 vector (3.1) ηn(p) = L
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Appendix B: Proofs for Section 5 Li
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Local asymptotic minimax risk/asymm
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Then (3.5) Local asymptotic minimax
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Moment-density estimation 323 may n
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3. The bias and MSE of ˆf ∗ α M
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Moment-density estimation 327 Corol
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Moment-density estimation 329 Suppo
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6. Simulations Moment-density estim
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Moment-density estimation 333 [7] M
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for positive α, and for α = 0 as
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Asymptotics of the MDPDE 337 Lemma
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Asymptotics of the MDPDE 339 asympt
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Optimality

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